Predicting quantum learnability from landscape fluctuation

The tradeoff between trainability and expressibility is a central challenge faced by today's variational quantum computing. Recent studies indicate that resolving this dilemma necessitates designing specific parametrized quantum circuits (PQC) tailored for specific problems, which urgently needs a general and efficient method to assess the learnability of PQCs regarding a given target. In this Letter, we demonstrate a simple and efficient metric for learnability by comparing the fluctuations of the given training landscape with standard learnable landscapes. This metric shows surprising effectiveness in predicting learnability as it unifies the effects of insufficient expressibility, barren plateaus, bad local minima, and overparametrization. Importantly, it does not require actual training and can be estimated efficiently on classical computers via Clifford sampling. We conduct extensive numerical experiments to validate its effectiveness regarding both physical and random Hamiltonians. We also prove a compact lower bound for the metric in locally scrambled circuits as analytical guidance. Our findings enable efficient predictions of learnability, allowing fast selection of suitable PQCs for a given problem without training, which can improve the efficiency of variational quantum computing especially when access to quantum devices is limited.